The Stacks project

Lemma 37.58.4. A composition of perfect morphisms of schemes is perfect.

Proof. This translates into the following algebra result: If $A \to B \to C$ are composable perfect ring maps then $A \to C$ is perfect. We have seen this is the case for pseudo-coherent in Lemma 37.57.4 and its proof. By assumption there exist integers $n$, $m$ such that $B$ has tor dimension $\leq n$ over $A$ and $C$ has tor dimension $\leq m$ over $B$. Then for any $A$-module $M$ we have

\[ M \otimes _ A^{\mathbf{L}} C = (M \otimes _ A^{\mathbf{L}} B) \otimes _ B^{\mathbf{L}} C \]

and the spectral sequence of More on Algebra, Example 15.62.4 shows that $\text{Tor}^ A_ p(M, C) = 0$ for $p > n + m$ as desired. $\square$


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